Subgroups — C₅×#13277 #11935997

Order 1280.

366 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#499	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#107	C₂	1	1	yes	maximal, characteristic
4	C₅×#1380	C₂	1	2	yes	maximal
5	C₅×#1380	C₂	1	2	yes	maximal
6	C₂×C₅×#125	—	2	1	no	—
7	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
8	C₅×#499	—	2	2	no	—
9	C₅×#499	C₄	1	4	yes	complemented
10	C₂×C₅×#107	C₂²	1	1	yes	characteristic
11	C₂×C₅×#107	C₂²	1	1	yes	characteristic
12	C₂×C₅×#107	C₂²	1	2	yes	—
13	C₂×C₅×#106	—	2	1	no	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₄×C₅×C₄⋊C₄	—	2	1	no	—
16	C₅×C₄⋊C₁₆	—	2	2	no	—
17	C₅×#340	—	2	1	no	—
18	#13277	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₅×#125	—	2	4	no	—
22	C₅×#125	—	4	2	no	—
23	C₂×C₅×C₄⋊C₄	—	2	2	no	—
24	C₂×C₅×C₄⋊C₄	—	2	1	no	—
25	C₂×C₅×C₄⋊C₄	—	2	1	no	—
26	C₂×C₅×C₄⋊C₄	—	4	1	no	—
27	C₂×C₅×C₄⋊C₄	—	4	1	no	—
28	C₂×C₄²×C₅	—	4	1	no	—
29	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
30	C₂×C₅×C₁₆	—	2	2	no	—
31	C₂×C₅×C₁₆	—	2	4	no	—
32	C₅×#107	—	2	1	no	—
33	C₅×#107	—	2	1	no	—
34	C₅×#107	—	2	2	no	—
35	C₅×#107	C₂×C₄	1	2	yes	—
36	C₅×#107	C₂×C₄	1	2	yes	—
37	C₅×#107	Q₈	1	4	yes	—
38	C₅×#106	—	2	4	no	—
39	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
40	C₄×C₅×C₈	D₄	1	2	yes	—
41	C₅×#95	—	4	1	no	—
42	C₂×#499	C₂×C₅	1	1	yes	characteristic
43	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic
44	C₄×#107	C₂×C₅	1	1	yes	characteristic
45	#1380	C₂×C₅	1	2	yes	—
46	#1380	C₂×C₅	1	2	yes	—
47	C₂²×C₄×C₅	—	4	2	no	—
48	C₂²×C₄×C₅	—	4	1	no	—
49	C₂²×C₄×C₅	—	4	1	no	—
50	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂×C₅×C₈	—	2	2	no	—
54	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
55	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
58	C₂×C₅×C₈	#41	1	2	yes	—
59	C₅×C₄⋊C₄	—	2	2	no	—
60	C₅×C₄⋊C₄	—	2	2	no	—
61	C₅×C₄⋊C₄	—	2	4	no	—
62	C₅×C₄⋊C₄	—	4	4	no	—
63	C₅×C₄⋊C₄	—	4	4	no	—
64	C₅×C₄⋊C₄	—	4	1	no	—
65	C₅×C₄⋊C₄	—	4	1	no	—
66	C₅×C₄⋊C₄	—	4	2	no	—
67	C₄²×C₅	—	8	2	no	—
68	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₅	SD₁₆	1	2	yes	—
70	C₅×C₁₆	—	4	2	no	—
71	C₂×#125	—	2	1	no	—
72	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
73	#499	—	2	2	no	—
74	#499	C₄×C₅	1	4	yes	complemented
75	C₂×#107	C₂²×C₅	1	1	yes	characteristic
76	C₂×#107	C₂²×C₅	1	1	yes	characteristic
77	C₂×#107	C₂²×C₅	1	2	yes	—
78	C₂×#106	—	2	1	no	—
79	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
80	C₄×C₄⋊C₄	—	2	1	no	—
81	C₄⋊C₁₆	—	2	2	no	—
82	#340	—	2	1	no	—
83	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	2	1	no	—
86	C₂×C₄×C₅	—	2	1	no	—
87	C₂×C₄×C₅	—	4	2	no	—
88	C₂×C₄×C₅	—	4	2	no	—
89	C₂×C₄×C₅	—	4	4	no	—
90	C₂×C₄×C₅	—	8	2	no	—
91	C₂×C₄×C₅	—	8	2	no	—
92	C₂×C₄×C₅	—	8	2	no	—
93	C₂×C₄×C₅	—	8	1	no	—
94	C₂×C₄×C₅	—	8	1	no	—
95	C₂×C₄×C₅	#102	1	2	yes	—
96	C₂×C₄×C₅	#102	1	2	yes	—
97	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
99	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅	#137	1	2	yes	—
101	C₅×C₈	—	2	2	no	—
102	C₅×C₈	C₄×Q₈	1	2	yes	—
103	C₅×C₈	#124	1	1	yes	characteristic
104	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
105	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
106	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
107	#125	—	2	4	no	—
108	#125	—	4	2	no	—
109	C₂×C₄⋊C₄	—	2	1	no	—
110	C₂×C₄⋊C₄	—	2	1	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	4	1	no	—
113	C₂×C₄⋊C₄	—	4	1	no	—
114	C₂×C₄²	—	4	1	no	—
115	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	2	4	no	—
118	#107	—	2	1	no	—
119	#107	—	2	1	no	—
120	#107	—	2	2	no	—
121	#107	C₂×C₄×C₅	1	2	yes	—
122	#107	C₂×C₄×C₅	1	2	yes	—
123	#107	C₅×Q₈	1	4	yes	—
124	#106	—	2	4	no	—
125	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
126	C₄×C₈	C₅×D₄	1	2	yes	—
127	#95	—	4	1	no	—
128	C₂²×C₅	C₂×#102	1	1	yes	characteristic
129	C₂²×C₅	#474	1	2	yes	—
130	C₂²×C₅	#508	1	2	yes	—
131	C₂²×C₅	#509	1	1	yes	characteristic
132	C₂²×C₅	#510	1	1	yes	characteristic
133	C₄×C₅	—	2	2	no	—
134	C₄×C₅	—	2	2	no	—
135	C₄×C₅	—	8	2	no	—
136	C₄×C₅	—	8	2	no	—
137	C₄×C₅	—	8	4	no	—
138	C₄×C₅	#391	1	1	yes	characteristic
139	C₄×C₅	#419	1	1	yes	characteristic
140	C₄×C₅	#475	1	2	yes	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	—	4	1	no	—
143	C₂²×C₄	—	4	1	no	—
144	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
145	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
149	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
150	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
151	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
152	C₂×C₈	C₅×#41	1	2	yes	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	1	no	—
157	C₄⋊C₄	—	4	1	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄²	—	8	2	no	—
162	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
163	C₄²	C₅×SD₁₆	1	2	yes	—
164	C₁₆	—	4	2	no	—
165	C₂×C₅	#1988	1	1	yes	characteristic
166	C₂×C₅	#2138	1	1	yes	characteristic
167	C₂×C₅	#2139	1	1	yes	characteristic
168	C₂×C₅	#2228	1	2	yes	—
169	C₂×C₅	#2229	1	2	yes	—
170	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	—	8	1	no	—
181	C₂×C₄	—	8	1	no	—
182	C₂×C₄	C₅×#102	1	2	yes	—
183	C₂×C₄	C₅×#102	1	2	yes	—
184	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
185	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
186	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
187	C₂×C₄	C₅×#137	1	2	yes	—
188	C₈	—	2	2	no	—
189	C₈	C₄×C₅×Q₈	1	2	yes	—
190	C₈	C₅×#124	1	1	yes	characteristic
191	C₈	C₅×C₄⋊Q₈	1	1	yes	characteristic
192	C₅	#13277	1	1	yes	characteristic, direct product factor, Sylow
193	C₂²	C₂×C₅×#102	1	1	yes	characteristic
194	C₂²	C₅×#474	1	2	yes	—
195	C₂²	C₅×#508	1	2	yes	—
196	C₂²	C₅×#509	1	1	yes	characteristic
197	C₂²	C₅×#510	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	8	2	no	—
201	C₄	—	8	2	no	—
202	C₄	—	8	4	no	—
203	C₄	C₅×#391	1	1	yes	characteristic
204	C₄	C₅×#419	1	1	yes	characteristic
205	C₄	C₅×#475	1	2	yes	—
206	C₂	C₅×#1988	1	1	yes	characteristic
207	C₂	C₅×#2138	1	1	yes	characteristic
208	C₂	C₅×#2139	1	1	yes	characteristic
209	C₂	C₅×#2228	1	2	yes	—
210	C₂	C₅×#2229	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
9. C₅×#499	C₄	semidirect
18. #13277	C₅	direct (N × H)
74. #499	C₄×C₅	semidirect
192. C₅	#13277	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#499	2
C₂×C₅×C₄⋊C₁₆	2
C₄×C₅×#107	2
C₅×#1380	2
#13277	5

Subgroups — C5×#13277 #11935997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13277 #11935997